Question

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ \left(3, \frac{\pi}{2}\right) $$

Short answer

(0, 3)

Step-by-step solution

Understand Polar Coordinates

Polar coordinates are given as \((r, \theta)\), where \(r\) is the radius and \(\theta\) is the angle measured from the positive x-axis. In this exercise, \(r = 3\) and \(\theta = \frac{\pi}{2}\).

Use Conversion Formulas

To convert polar coordinates to rectangular coordinates, use the formulas: \(x = r \cdot \cos(\theta)\) and \(y = r \cdot \sin(\theta)\).

Calculate the x-Coordinate

Substitute \(r = 3\) and \(\theta = \frac{\pi}{2}\) into the x-coordinate formula: \( x = 3 \cdot \cos\left(\frac{\pi}{2}\right) = 3 \cdot 0 = 0 \). Hence, the x-coordinate is 0.

Calculate the y-Coordinate

Substitute \(r = 3\) and \(\theta = \frac{\pi}{2}\) into the y-coordinate formula: \( y = 3 \cdot \sin\left(\frac{\pi}{2}\right) = 3 \cdot 1 = 3 \). Hence, the y-coordinate is 3.

Write the Rectangular Coordinates

After calculating both coordinates, combine them to form the rectangular coordinates: \( (x, y) = (0, 3) \).

Key concepts

Polar Coordinates

Polar coordinates are an alternative way of describing a point in a plane. Instead of using horizontal and vertical distances, polar coordinates use a radius and an angle.
The radius, denoted by \( r \), is the distance from the origin to the point. The angle, denoted by \( \theta \), is the measure of the counter-clockwise angle from the positive x-axis to the point. This method is particularly useful in scenarios involving circular or rotational symmetry.
For example, in our problem, the given polar coordinates are \( (3, \frac{\pi}{2}) \), meaning the point is 3 units away from the origin and forms an angle of \( \frac{\pi}{2} \) radians (90 degrees) with the positive x-axis.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a point in a plane using two perpendicular lines (the x-axis and the y-axis). Each point is characterized by an ordered pair \( (x, y) \), where \( x \) represents the horizontal distance from the origin and \( y \) represents the vertical distance.
In our exercise, we need to convert the polar coordinates \( (3, \frac{\pi}{2}) \) into rectangular coordinates. This involves calculating the exact x and y values using conversion formulas derived from trigonometry.

Trigonometric Functions

Trigonometric functions help us relate angles to side lengths in right triangles. The main trigonometric functions used in coordinate conversions are sine \( \sin \) and cosine \( \cos \).
The cosine function, \( \cos(\theta) \), gives the ratio of the adjacent side to the hypotenuse in a right triangle. The sine function, \( \sin(\theta) \), gives the ratio of the opposite side to the hypotenuse.
When converting from polar to rectangular coordinates, we use these trigonometric functions as follows:

• \( x = r \cdot \cos(\theta) \)
• \( y = r \cdot \sin(\theta) \)

This relationship allows us to translate the radial distance and angular information into horizontal and vertical distances.

Coordinate Geometry

Coordinate geometry is a system that uses coordinates to represent geometrical shapes and analyze their properties. With the Cartesian coordinate system, we can precisely locate points and describe geometric figures using equations.
In our example, we converted the polar coordinates \( (3, \frac{\pi}{2}) \) to the rectangular coordinates \( (0, 3) \) as follows:
Using the formulas:

• \( x = 3 \cdot \cos(\frac{\pi}{2}) = 0 \)
• \( y = 3 \cdot \sin(\frac{\pi}{2}) = 3 \)

So, the point (3, 90 degrees) in polar coordinates translates to the point (0, 3) on the Cartesian plane. This helps us understand the exact position of the point using straightforward x and y values, making further mathematical manipulations or visualizations easier.